All Questions
Tagged with linear-algebrametric-tensor
49 questions
3votes
2answers
285views
The inverse of a specific metric tensor [closed]
I am studying general relativity and here is a problem I encountered: Suppose $$ \mathrm{d}s^2=-M^2(\mathrm{d}t-M_i\mathrm{d}x^i)(\mathrm{d}t-M_j\mathrm{d}x^j)+g_{ij}\mathrm{d}x^i\mathrm{d}x^j $$ or ...
- 129
1vote
1answer
132views
Exponential of the metric tensor
Exponential of an arbitrary matrix can be written as $$e^A = \displaystyle\sum_{n=0}^\infty \dfrac{A^n}{n!}$$ In Einstein notation, how this expression will look like? In Einstein notation, what ...
- 1,177
-1votes
2answers
108views
Is it possible for a Lorentz scalar to NOT be invariant under another linear transformation?
Lorentz scalars are invariant under Lorentz transformations, which are a subset of linear transformations. I wanted to know if it is possible, for a Lorentz scalar, to NOT be invariant with respect to ...
- 484
0votes
1answer
125views
Transformations that preserve the metric [duplicate]
I know that transformations that preserve the metric (like the Lorentz transformation, or rotations) have the property: $$S^T \eta S = \eta$$ However, I'm getting: $S^TS = I$ and I'm not sure why: $$\...
- 1,376
0votes
0answers
674views
How to know if a matrix is a (0,2) tensor, a (2,0) tensor or a (1,1) tensor?
For example: $$ X = \begin{bmatrix} 1 & -1 & 0 & 0 \\ -1 & 0 & 5 & 3 \\ -2 & 1 & 0 & 0 \\ 0 & 1 & 0 & 2 ...
0votes
2answers
61views
Could you explain why there is a transpose in this proof? (Invariance of distances with linear transformation)
The equation is supposed to be $$s^2 = \eta_{\mu \nu} \Delta x^\mu \Delta x^\nu.$$ How is it transformed to, $$s^2 = (\Delta x)^T\eta (\Delta x) = (\Delta x')^T\eta (\Delta x')~? $$ Why is there a ...
4votes
2answers
710views
Confusion on metric determinant derivative
Maybe it is a stupid confusion. I need to compute the derivative of the metric determinant with respect to the metric itself, i.e., $\partial g/\partial g_{\mu\nu}$, but I have an indices confusion in ...
3votes
2answers
374views
Problem with proving the invariance of dot product of two four vectors
I am having a spot of trouble with index manipulation (its not that I am very unfamiliar with this, but I keep losing touch). This is from an electrodynamics course - we're just getting started with 4 ...
- 322
4votes
1answer
544views
Notion of Co- and Contravariance in Dirac-Notation
$\newcommand{\bra}[1]{\left<#1\right|}\newcommand{\ket}[1]{\left|#1\right>}\newcommand{\bk}[2]{\left<#1\middle|#2\right>}\newcommand{\bke}[3]{\left<#1\middle|#2\middle|#3\right>}$ A (...
- 406
0votes
1answer
87views
Use Index Notation properly when indices are already used in identifying which bases is the matrix metric calculated with respect to
I am wondering how to apply the usual linear algebra to the rather unfamiliar case of 'matrices' with indices in special relativity or even general relativity. In particular, consider$$f=\sqrt{-\det\...
- 942
0votes
1answer
661views
Determinant and inverse of metric tensor in Eddington-Finkelstein coordinates
I need to make sure I'm not going crazy here. We define the metric of Eddington-Finkelstein (EF) coordinates $(v,r, \theta,\phi)$ as $$g=-\left(1-\frac{2m}{r}\right)dv^2+2dvdr+r^2d\Omega^2$$ where $d\...
user344261
1vote
1answer
937views
Null vectors that aren't the zero vector in general relativity?
So I was trying to understand the null energy condition of $T_{μν}k^μk^ν≥0$ Where $k$ is an "arbitrary future-directed null vector" and couldn't really wrap my head around how the $k$ is ...
1vote
1answer
1kviews
Why can the metric tensor always be diagonalized?
I'm reading through some general relativity notes. I have reached a part that I don't understand, probably because my linear algebra is not good enough. My questions relating to the image below are: ...
- 947
1vote
1answer
89views
Weyl, Space-Time-Matter: The Riemann "quadrilinear form", and "functions which stand in quadratic relationship with an element of surface"
The following is from pages 83 and 84 of Space—Time—Matter by Hermann Weyl: Sometimes conditions of symmetry more complicated than those considered heretofore occur. In the realm of quadrilinear ...
- 2,086
4votes
3answers
2kviews
How to multiply matrices (the physicists way) using index notation?
N.B For this question we are only working in in flat cartesian space, not curved space-time. Question: Consider $A$, $B$, $C$ and $D$ to be $n \times n$ matrices. Write down the following matrix ...
user267061
